Books, Proceedings and Seminars of Centre Mersenne

[1] L. A. Ambrosio, N. Gigli, and G. Savaré, Gradient flows in metric spaces and in the space of probability measures, Lectures in Mathematics, Birkhäuser, 2005. | MR | Zbl

[2] J. Bedrossian and Inwon Kim, Global Existence and Finite Time Blow-Up for Critical Patlak-Keller-Segel Models with Inhomogeneous, preprint, 2011.

[3] J. Bedrossian, N. Rodríguez and A. Bertozzi, Local and global well-posedness for aggregation equations and Patlak-Keller-Segel models with degenerate diffusion, Nonlinearity, 24 (2011), pp. 1683–1715. | MR | Zbl

[4] A. L. Bertozzi and M. C. Pugh, Long-wave instabilities and saturation in thin film equations, Comm. Pure Appl. Math., 51 (1998), pp. 625–661. | MR | Zbl

[5] Idem, Finite-time blow-up of solutions of some long-wave unstable thin film equations, Indiana Univ. Math. J., 49 (2000), pp. 1323–1366. | MR

[6] P. Biler, L. Corrias and J. Dolbeault, Large mass self-similar solutions of the parabolic-parabolic Keller-Segel model of chemotaxis, J. of Math. Biol., 63 (2011), pp. 1–32. | MR | Zbl

[7] P. Biler, G. Karch, P. Laurençot and T. Nadzieja, The $8\pi$-problem for radially symmetric solutions of a chemotaxis model in the plane, Math. Methods Appl. Sci., 29 (2006), pp. 1563–1583. | MR | Zbl

[8] A. Blanchet, V. Calvez and J.A. Carrillo, Convergence of the mass-transport steepest descent scheme for the subcritical Patlak-Keller-Segel model, SIAM J. Numer. Anal., 46 (2008), pp. 691–721. | MR | Zbl

[9] A. Blanchet, E. Carlen and J.A. Carrillo, Functional inequalities, thick tails and asymptotics for the critical mass Patlak-Keller-Segel model, preprint, arXiv:1009.0134. | MR | Zbl

[10] A. Blanchet, J.A. Carrillo and Ph. Laurençot, Critical mass for a Patlak-Keller-Segel model with degenerate diffusion in higher dimensions, Calc. Var. Partial Differential Equations, 35 (2009), pp. 133–168. | MR | Zbl

[11] A. Blanchet, J. A. Carrillo and N. Masmoudi, Infinite Time Aggregation for the Critical Patlak-Keller-Segel model in ${ℝ}^2$, Comm. Pure Appl. Math., 61 (2008), pp. 1449–1481. | MR | Zbl

[12] A. Blanchet, J. Dolbeault, M. Escobedo and J. Fernandez, Asymptotic behaviour for small mass in the two-dimensional parabolic-elliptic Keller-Segel model, Journal of Mathematical Analysis and Applications, 361 (2010), pp. 533–542.. | MR | Zbl

[13] A. Blanchet, J. Dolbeault and B. Perthame, Two-dimensional Keller-Segel model: optimal critical mass and qualitative properties of the solutions, Electron. J. Differential Equations, 44 (2006), 32 pp. (electronic). | MR | Zbl

[14] A. Blanchet and Ph. Laurençot, Finite mass self-similar blowing-up solutions of a chemotaxis system with non-linear diffusion, To appear in Communications on Pure and Applied Analysis (2010). | MR

[15] M. P. Brenner, L. S. Levitov and E. O. Budrene, Physical Mechanisms for Chemotactic Pattern Formation by Bacteria, Biophysical Journal 74 (1998), pp. 1677–1693.

[16] M. Burger, Y. Dolak-Struss, and C. Schmeiser, Asymptotic analysis of an advection-dominated chemotaxis model in multiple spatial dimensions, Commun. Math. Sci., 6 (2008), pp. 1–28. | MR | Zbl

[17] V. Calvez and J. A. Carrillo, Volume effects in the Keller-Segel model: energy estimates preventing blow-up, J. Math. Pure et Appl., 86 (2006), pp. 155–175. | MR | Zbl

[18] V. Calvez and L. Corrias, The parabolic-parabolic Keller-Segel model in ${ℝ}^2$, Commun. Math. Sci. 6 (2008), pp. 417–447. | MR | Zbl

[19] V. Calvez, R. Hawkins, N. Meunier and R. Voituriez, Analysis of a non local model for spontaneous cell polarisation, arXiv:1105.4429.

[20] E. Carlen and A. Figalli, Stability for a GNS inequality and the Log-HLS inequality, with application to the critical mass Keller-Segel equation, preprint, arXiv:1107.5976.

[21] J. A. Carrillo, R. J. McCann and C. Villani, Kinetic equilibration rates for granular media and related equations: entropy dissipation and mass transportation estimates, Rev. Matemática Iberoamericana, 19 (2003), pp. 1–48. | MR | Zbl

[22] T. Cazenave, Semilinear Schrödinger equations, Courant Lecture Notes in Mathematics, 10, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003. | MR | Zbl

[23] P.H. Chavanis, Generalized thermodynamics and Fokker-Planck equations. Applications to stellar dynamics, two-dimensional turbulence and Jupiter’s great red spot, Phys. Rev. E, 68 (2003), pp. 036108.

[24] P.H. Chavanis, Nonlinear mean field Fokker-Planck equations. Application to the chemotaxis of biological populations, Eur. Phys. J. B, 62 (2008), pp. 179–208 . | Zbl

[25] P.-H. Chavanis and R. Mannella, Self-gravitating Brownian particles in two dimensions: the case of $N=2$ particles, The Eur. Phys. J. B, 78 (2010), pp. 139–165. | MR

[26] P.-H. Chavanis and C. Sire, Anomalous diffusion and collapse of self-gravitating Langevin particles in $D$ dimensions, Phys. Rev. E, 69 (2004), 016116.

[27] S. Childress, Chemotactic collapse in two dimensions, Lecture Notes in Biomath, 55 (1984), pp. 217–237. | MR

[28] S. Childress and J. K. Percus, Nonlinear aspects of chemotaxis, Math. Biosci., 56 (1981), pp. 217–237. | MR | Zbl

[29] L. Corrias, B. Perthame, and H. Zaag, Global solutions of some chemotaxis and angiogenesis systems in high space dimensions, Milan J. Math., 72 (2004), pp. 1–29. | MR | Zbl

[30] M. Del Pino and J. Dolbeault, Best constants for Gagliardo-Nirenberg inequalities and applications to nonlinear diffusions, J. Math. Pures Appl., 81 (2002), pp. 847–875. | MR | Zbl

[31] J. Dolbeault and B. Perthame, Optimal critical mass in the two-dimensional Keller-Segel model in ${ℝ}^2$, C. R. Math. Acad. Sci. Paris, 339 (2004), pp. 611–616. | MR | Zbl

[32] J. Dolbeault and C. Schmeiser, The two-dimensional Keller-Segel model after blow-up, Disc. Cont. Dynam. Systems B, 25 (2009), pp. 109–121. | MR | Zbl

[33] M. A. Herrero and J. J. L. Velázquez, Chemotactic collapse for the Keller-Segel model, J. of Math. Biol., 35 (1996), pp. 177–194. | MR | Zbl

[34] M. A. Herrero and J. J. L. Velázquez, Singularity patterns in a chemotaxis model, Math. Ann., 306 (1996), pp. 583–623. | MR | Zbl

[35] T. Hillen and K. Painter, Global existence for a parabolic chemotaxis model with prevention of overcrowding, Adv. in Appl. Math., 26 (2001), pp. 280–301. | MR | Zbl

[36] T. Hillen and K. Painter, A user’s guide to PDE models for chemotaxis, Journal of Mathematical Biology, 58 (2009), pp. 183–217. | MR | Zbl

[37] D. Horstmann, From 1970 until present: the Keller-Segel model in chemotaxis and its consequences. I, Jahresber. Deutsch. Math.-Verein., 105 (2003), pp. 103–165. | MR | Zbl

[38] Idem, From 1970 until present: the Keller-Segel model in chemotaxis and its consequences. II, Jahresber. Deutsch. Math.-Verein., 106 (2004), pp. 51–69. | MR | Zbl

[39] W. Jäger and S. Luckhaus, On explosions of solutions to a system of partial differential equations modelling chemotaxis, Trans. Amer. Math. Soc., 329 (1992), pp. 819–824. | MR | Zbl

[40] N. Kavallaris and P. Souplet, Grow-up rate and refined asymptotics for a two-dimensional Patlak-Keller-Segel model in a disk, SIAM J. Math. Anal., 40 (2008/09), pp. 1852–1881. | MR

[41] E. F. Keller and L. A. Segel, Initiation of slide mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), pp. 399–415. | Zbl

[42] R. Kowalczyk, Preventing blow-up in a chemotaxis model, J. Math. Anal. Appl. 305 (2005), pp. 566–588. | MR | Zbl

[43] C. Lederman and P. A. Markowich, On fast-diffusion equations with infinite equilibrium entropy and finite equilibrium mass, Comm. Partial Differential Equations, 28 (2003), pp. 301–332. | MR | Zbl

[44] S. Luckhaus, Y. Sugiyama, J. J. L. Velázquez, Measure valued solutions of the 2D Keller-Segel system, arXiv:1011.0282. | MR

[45] P. M. Lushnikov, Critical chemotactic collapse, Phys. Lett. A 374 (2010), pp. 1678–1685. | Zbl

[46] M. A. Herrero, E. Medina and J. J. L. Velázquez, Self-similar blow-up for a reaction-diffusion system, J. of Comp. and Appl. Math., 97 (1998), pp. 99–119. | MR | Zbl

[47] F. Merle and P. Raphaël, On universality of blow-up profile for $L^2$ critical nonlinear Schrödinger equation, Invent. Math., 156 (2004), pp. 565–672. | MR | Zbl

[48] T. Nagai, Blow-up of radially symmetric solutions to a chemotaxis system, Adv. Math. Sci. Appl., 5 (1995), pp. 581–601. | MR | Zbl

[49] T. Nagai, T. Senba, T. Suzuki, Chemotactic collapse in a parabolic system of mathematical biology, Hiroshima Math. J, 30 (2000), pp. 463–497. | MR | Zbl

[50] V. Nanjundiah, Chemotaxis, signal relaying and aggregation morphology, Journal of Theoretical Biology, 42 (1973), pp. 63–105.

[51] C. S. Patlak, Random walk with persistence and external bias, Bull. Math. Biophys., 15 (1953), pp. 311–338. | MR

[52] B. Perthame, Transport equations in biology, Frontiers in Mathematics, Birkhäuser Verlag, Basel, 2007. | MR | Zbl

[53] D. Slepčev and M. C. Pugh, Selfsimilar blowup of unstable thin-film equations, Indiana Univ. Math. J., 54 (2005), pp. 1697–1738. | MR | Zbl

[54] C. Sire and P.-H. Chavanis, Gravitational collapse of a Brownian gas, Banach Center Publ. 66, 287, 2004. | MR | Zbl

[55] C. Sire and P.-H. Chavanis, Critical dynamics of self-gravitating Langevin particles and bacterial populations, Phys. Rev. E, 78 (2008), 061111. | MR

[56] Y. Sugiyama, Global existence in sub-critical cases and finite time blow-up in super-critical cases to degenerate Keller-Segel systems, Diff. Int. Eq., 19 (2006), pp. 841–876. | MR | Zbl

[57] Y. Sugiyama, Application of the best constant of the Sobolev inequality to degenerate Keller-Segel models, Adv. Diff. Eq., 12 (2007), pp. 121–144. | MR | Zbl

[58] Y. Sugiyama and J. J. L. Velázquez, Self-similar blow-up with a continuous range of values of the aggregated mass for a degenerate Keller-Segel system, Adv. Diff. Eq., 16 (2011), pp. 85–112. | MR | Zbl

[59] C. Sulem and P. L. Sulem, The nonlinear Schrödinger equation, Applied Mathematical Sciences 139, Springer-Verlag, New York, 1999. | MR | Zbl

[60] T. Suzuki and R. Takahashi, Degenerate parabolic equation with critical exponent derived from the kinetic theory, I, generation of the weak solution, Adv. Diff. Eq., 14 (2009), pp. 433–476. | MR | Zbl

[61] J. J. L. Velázquez, Stability of some mechanisms of chemotactic aggregation, SIAM J. Appl. Math., 62 (2002), pp. 1581–1633 (electronic). | MR | Zbl

[62] Idem, Point dynamics in a singular limit of the Keller-Segel model. I. Motion of the concentration regions, SIAM J. Appl. Math., 64 (2004), pp. 1198–1223 (electronic). | MR | Zbl

[63] Idem, Point dynamics in a singular limit of the Keller-Segel model. II. Formation of the concentration regions, SIAM J. Appl. Math., 64 (2004), pp. 1224–1248 (electronic). | MR | Zbl

[64] C. Villani, Topics in optimal transportation, Graduate Studies in Mathematics Vol. 58, Amer. Math. Soc, Providence, 2003. | MR | Zbl

[65] M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys. 87 (1983), pp. 567–576. | MR | Zbl